Now we give a description of the rarefaction wave connected to the
vacuum to the compressible Euler equations (REF ), see also the
references [1]} and [2]}. For definiteness,
2-rarefaction wave will be considered. If we investigate the
compressible
Euler system (REF ) with the Riemann initial data
\(\left\lbrace \begin{array}{rr} \rho (0,x)=0,&x<0,\\(\rho ,u)(0,x)=(\rho _+,u_+),&x>0,\end{array}\right.\) 
